Geometry & Topology, Vol. 7 (2003)
Paper no. 9, pages 311--319.

On Invariants of Hirzebruch and Cheeger-Gromov

Stanley Chang, Shmuel Weinberger

Abstract.
We prove that, if M is a compact oriented manifold of dimension 4k+3,
where k>0, such that pi_1(M) is not torsion-free, then there are
infinitely many manifolds that are homotopic equivalent to M but not
homeomorphic to it. To show the infinite size of the structure set of
M, we construct a secondary invariant tau_(2): S(M)-->R that coincides
with the rho-invariant of Cheeger-Gromov. In particular, our result
shows that the rho-invariant is not a homotopy invariant for the
manifolds in question.